On Physical Lines of Force/Part II/Prop. VIII

On Physical Lines of Force → Part II → Prop. VIII

Maxwell's statement

Prop. VIII. — To find the relations between the alterations of motion of the vortices, and the forces P, Q, R which they exert on the layer of particles between them.

This proposition produces Faraday's law of induction, the vector potential, and the statement that there are no magnetic monopoles — five numbered equations, (54) to (58), in a few pages.

The comparison

Let V be the volume of a vortex, then by (46) its energy is …

Differentiate that with respect to time. Then:

Comparing this value with that given in equation (50) …

The two must be equal, because the only source of the vortex's energy is the layer of particles. The resulting identity must hold whatever $\alpha ,\beta$ and $\gamma$ do — they are independent — so the coefficient of each may be separated out on its own. Three equations result.

From these equations we may determine the relation between the alterations of motion [of the vortices] and the forces exerted on the layers of particles between the vortices, or, in the language of our hypothesis, the relation between changes in the state of the magnetic field and the electromotive forces thereby brought into play.

Equation (54): Faraday's law

$$\frac{d(\mu \alpha )}{dt}=\frac{dQ}{dz}-\frac{dR}{dy},\frac{d(\mu \beta )}{dt}=\frac{dR}{dx}-\frac{dP}{dz},\frac{d(\mu \gamma )}{dt}=\frac{dP}{dy}-\frac{dQ}{dx}\mathrm{(}\mathrm{54}\mathrm{)}$$

Provenance: the number (54) is attested and its content — the relation between changes in the magnetic field and the electromotive forces brought into play — is fixed by Maxwell's own gloss above. The body is a reconstruction. See Sources.

In modern notation:

$$\mathrm{\nabla }\times 𝐄=-\frac{\partial 𝐁}{\partial t}$$

This is Faraday's law of induction, obtained from the conservation of energy in a system of flywheels. Faraday had found it in the laboratory in 1831 as a rule about circuits and changing flux. Here it appears as a local statement, true at every point of space, and derived rather than assumed.

The minus sign — Lenz's law, the fact that induction opposes the change producing it — is not put in. It comes from the inertia of the vortices: a spinning mass resists being spun faster, and its resistance is what the particles feel. See Induced currents for the physical narration.

Equation (55): the electrotonic state

Maxwell now asks what quantity, differentiated, gives the magnetic force. He introduces $F,G,H$:

$$\mu \alpha =\frac{dH}{dy}-\frac{dG}{dz},\mu \beta =\frac{dF}{dz}-\frac{dH}{dx},\mu \gamma =\frac{dG}{dx}-\frac{dF}{dy}\mathrm{(}\mathrm{55}\mathrm{)}$$

In modern notation $𝐁=\mathrm{\nabla }\times 𝐀$. $F,G,H$ are the components of the vector potential — and Maxwell identifies them with Faraday's "electrotonic state", the condition of a body under induction that Faraday had posited without being able to say what it was. Maxwell had already given the electrotonic state a geometrical meaning in his 1855–56 paper; here it acquires a mechanical one, as the electromagnetic momentum of the medium.

On the word "momentum". The analogy Maxwell is drawing is exact and deliberate. In ordinary mechanics, force is the rate of change of momentum. In equation (58) below, electromotive force is the rate of change of $F,G,H$. Whatever $F,G,H$ are, they play the part momentum plays — which is why he calls the electrotonic state the electromagnetic momentum, and why the quantity survives into modern physics as the vector potential.

Equation (56): no magnetic monopoles

Taking the divergence of (55) gives identically zero, since the divergence of a curl vanishes:

$$\frac{d(\mu \alpha )}{dx}+\frac{d(\mu \beta )}{dy}+\frac{d(\mu \gamma )}{dz}=0\mathrm{(}\mathrm{56}\mathrm{)}$$

In modern notation $\mathrm{\nabla }\cdot 𝐁=0$. Magnetic lines of force have no ends. On the vortex hypothesis this is not a separate postulate at all: a vortex is a closed circulation, and a circulation has no source.

Note the tension with Part I, where the first term of equations (12) was the force on a magnetic pole, expressed through "imaginary magnetic matter" with a non-zero divergence. Maxwell uses the pole language as a computational device while denying that poles are real sources — a distinction still made today.

Equations (57) and (58): electromotive force

Differentiating (55) with respect to time and comparing with (54) — equation (57) — shows that $P,Q,R$ and the time derivatives of $F,G,H$ can differ only by the gradient of some scalar. Calling that scalar $\psi$, the electric tension, gives equation (58):

$$P=-\frac{dF}{dt}-\frac{d\psi }{dx},Q=-\frac{dG}{dt}-\frac{d\psi }{dy},R=-\frac{dH}{dt}-\frac{d\psi }{dz}\mathrm{(}\mathrm{58}\mathrm{)}$$

Provenance: the numbers (55), (56), (57), (58) are attested; the bodies are reconstructions in Maxwell's notation. See Sources.

In modern notation:

$$𝐄=-\frac{\partial 𝐀}{\partial t}-\mathrm{\nabla }\phi$$

which is how the electric field is written in terms of potentials in every textbook since. Two sources of electromotive force, mechanically distinct on Maxwell's model:

• $-d𝐀/dt$ — the medium's electromagnetic momentum changing. This is induction.
• $-\mathrm{\nabla }\psi$ — the particles pressing on one another. This is electrostatics.

Summary of Prop. VIII

equation	Maxwell	modern	what it says
(54)	$d(\mu \alpha )/dt=dQ/dz-dR/dy$	$\mathrm{\nabla }\times 𝐄=-\partial 𝐁/\partial t$	changing magnetic field drives electricity
(55)	$\mu \alpha =dH/dy-dG/dz$	$𝐁=\mathrm{\nabla }\times 𝐀$	the electrotonic state is the vector potential
(56)	$d(\mu \alpha )/dx+\dots =0$	$\mathrm{\nabla }\cdot 𝐁=0$	no magnetic sources
(58)	$P=-dF/dt-d\psi /dx$	$𝐄=-\partial 𝐀/\partial t-\mathrm{\nabla }\phi$	the two sources of electromotive force

Three of the four Maxwell equations are now in hand. The fourth — Gauss's law for electricity — and the correction to the first await Part III.

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Next: Props. IX and X · Previous: Prop. VII

Modern derivations: Faraday's law · The vector potential · No magnetic monopoles